Killing tensors and canonical geometry

0.16 MB·

by  M. Cariglia

arXiv:1401.8195v3
Killing tensors and canonical geometry
M. Cariglia1 G. W.Gibbons2 , J.-W. van Holten3,4 ,
∗ † ‡
P. A. Horvathy5,6 , P. Kosin´ski7 , P.-M. Zhang5,8
§ ¶ ∗∗
1DEFIS, Universidade Federal de Ouro Preto,
Campus Moro de Cruzeiro, 35400-000 Ouro Preto, MG-Brasil
2Department of Applied Mathematics and Theoretical Physics,
4
1
0 Cambridge University, Cambridge, UK
2
3NIKHEF, Amsterdam (Netherlands)
r
p
A 4 Leiden University, Leiden (Netherlands)

5 5Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou (China)
2
6Laboratoire de Math´ematiques et de Physique Th´eorique, Tours University (France)
]
h
t 7Faculty of Physics and Applied Informatics, University of Lodz, (Poland)
-
p
e 8 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
h
[ Chinese Academy of Sciences, Beijing 100190, China

3
v (Dated: April 28, 2014)
5
9 Abstract
1
8
The systematic derivation of constants of the motion, based on Killing tensors and the gauge
.
1
0 covariant approach, is outlined. Quantum dots are shown to support second-, third- and fourth-
4
1
rank Killing tensors.
:
v
i KEY WORDS: Conserved quantities, Killing tensors, Covariant Dynamics, Quantum Dots
X
r
a
PACS numbers:
45.05.+x,General theory of classical mechanics of discrete systems
11.30.Na, Nonlinear and dynamical symmetries (spectrum-generating symmetries)
73.21.La,Quantum dots
∗ e-mail:[email protected]
† mail:[email protected]
‡ mail: [email protected]
§ e-mail:horvathy-at-lmpt.univ-tours.fr
¶ email: [email protected]
∗∗ e-mail:[email protected]
1
Contents
1. Introduction 2
2. Canonical approach to conservation laws 3
3. Application to Quantum Dots 4
4. Algebraic Structure 10
5. Conclusion 11
Acknowledgments 12
References 12
1. INTRODUCTION
Noether’s theorem associates a conserved quantity to a symmetry, defined as a transfor-
mation of space-time which changes the Lagrangian by a total derivative [1]. Infinitesimally,
such symmetries are Killing vectors. Higher-order expressions like the celebrated (Laplace-)
Runge-Lenz vector cannot be obtained in this way, though, and require rather higher-rank
generalizations called Killing tensors [2–7]. The relations between these concepts become
particularly clear in the canonical formulation of the dynamics [8, 9].
The (Laplace-) Runge-Lenz vector is associated with rank-2 tensors. Killing tensors of
rank r ≥ 2 can also be considered, but physical examples are less common [10–15].
The main result of this paper is a systematic derivation of higher-rank conserved quan-
tities based on Killing tensors, as illustrated by a fourth-rank conserved quantity, (3.24)
below. It is obtained for Quantum Dots for a particular choice of the parameters, when the
system is integrable but not separable [16–19]. Deriving this expression is far from being
trivial : Blu¨mel et al. [18], e.g., found the correct formula at their second attempt only (and
gave no detailed explanation). The difficulty comes from that such higher-order expressions
are, as said above, not associated with a simple geometric action onspace-time and therefore
cannot be derived by the original Noether theorem.
2
2. CANONICAL APPROACH TO CONSERVATION LAWS
For particles coupled to scalar and vector potentials (Φ,A ) the hamiltonian takes the
i
form
1
H = gij(q)Π Π +Φ(q), (2.1)
i j
2
where the Π = p −eA denote the gauge-covariant momenta; e is the charge. The covariant
i i i
brackets read [8, 9]
∂K ∂Q ∂Q ∂K
{Q,K} = D Q − D K +eF (q) , (2.2)
i i ij
∂Π ∂Π ∂Π ∂Π
i i i j
where F = ∂ A −∂ A is the field-strength tensor of A on the configuration space with
ij i j j i i
metric g , and the covariant derivatives are defined by
ij
∂K ∂K
D K ≡ +Γ kΠ . (2.3)
i ∂qi ij k ∂Π
(cid:12)Π j
(cid:12)
The definition reproduces the canonical equ(cid:12)ations of motion:
(cid:12)
dqi ∂H dqj
= qi,H = ⇔ Π = g , (2.4)
i ij
dt ∂Π dt
i
dΠ (cid:8) (cid:9) ∂H
i
= {Π ,H} = −D H +eF
i i ij
dt ∂Π
j (2.5)
DΠ dΠ dqj ∂Φ
⇔ i ≡ i − Γ kΠ = eF jΠ − .
Dt dt dt ji k i j ∂qi
Constants of the motion which are polynomials in the covariant momenta are obtained
by solving eq.
1
{Q,H} = 0 with Q = C(n)i1....in(q)Π ...Π , (2.6)
n! i1 in
n 0
X≥
which leads to the generalized Killing equations
C(1)iΦ = 0, C(0) = eC(1)jF +C(2)jΦ ,
;i ;i ij i ;j
(2.7)
(n) (n+1)j (n+2)j
C = eC F +C Φ , n ≥ 1,
(i1..in;in+1) (i1...in in+1)j i1...in+1 ;j
as discussed in Ref. [9]. Note that any Q which is a polynomial in Π of rank n, has C(m) = 0
i
for all m > n. In that case the highest coefficient tensor C(n) is a again a rank-n Killing
tensor:
(n)
C = 0; (2.8)
(i1...in;in+1)
the next-to-highest coefficient tensor satisfies
(n 1) (n)j
C(i1−...in−1;in) = eC(i1...in−1Fin)j, (2.9)
3
and all tensors of rank n−2 and lower are subject to the full equation (2.7). Furthermore,
the generalized Killing vector C(1) is always required to be orthogonal to the gradient of the
scalar potential.
3. APPLICATION TO QUANTUM DOTS
The above formalism can be applied to the Quantum-Dot model of ref. [16–19]. This
concerns two charged particles with Coulomb interaction in a constant magnetic field and a
confining oscillator potential. The hamiltonian is
2
1 a
H = Π2 +U(r ) − . (3.1)
2 a a |r −r |
a=1 (cid:20) (cid:21) 1 2
X
The magnetic field direction is the z-direction, and the confining oscillator potential is taken
to be axially symmetric:
1
U(r ) = ω2(x2 +y2)+ω2z2 . (3.2)
a 2 0 a a z a
Transformationtocenter-of-massco-ordi(cid:2)natesr = 1 (R±(cid:3)r), Π = 1 (Π±π),leads
1,2 √2 1,2 √2
to separation of variables, H(r ,Πi ) = H (R,Π)+H (r,π), with H = 1Π2+U(R)
a a CM red CM 2
and
1 a
H = π2 +U(r)− . (3.3)
red
2 r
As the magnetic field is constant, F = −F = B, F = F = 0, the covariant brackets
xy yx yz zx
separate as well: {Q,K} = {Q,K} +{Q,K} . In euclidean co-ordinates they read
CM red
∂K ∂Q ∂K ∂Q ∂K ∂Q ∂K ∂K
{K,Q} = − +eB − ,
CM ∂Ri ∂Π ∂Π ∂Ri ∂Π ∂Π ∂Π ∂Π
i i (cid:18) x y y x(cid:19)
(3.4)
∂K ∂Q ∂K ∂Q ∂K ∂Q ∂K ∂K
{K,Q} = − +eB − .
red ∂ri ∂π ∂π ∂ri ∂π ∂π ∂π ∂π
i i (cid:18) x y y x(cid:19)
In the following we restrict ourselves to the effective 1-particle problem defined by the
reduced brackets and hamiltonian H . To apply the formalism of generalized Killing equa-
red
tions we lump the oscillator and Coulomb potential into the single scalar potential
1 a
Φ = ω2(x2 +y2)+ω2z2 − . (3.5)
2 0 z x2 +y2+z2
(cid:2) (cid:3)
The equations of motion in 3D euclidean co-ordinatespderived from the reduced hamiltonian
and brackets then read
r˙ = {r ,H} = π , π˙ = {π ,H} = −Φ +ǫ eBπ , (3.6)
i i i i i ,i ijz j
4
where the comma denotes an ordinary partial derivative w.r.t. r , and ǫ is the permutation
i ijk
tensor. The corresponding quantum theory is obtained by replacing the brackets of phase-
space functions (K,Q) by operator commutation relations.
To make use of axial symmetry, it is convenient to transform to curvilinear co-ordinates
ξi = (ρ,z,ϕ). Then the hamiltonian becomes
1
H = gijπ π +Φ, (3.7)
i j
2
with metric g = diag(1,1,ρ2) and scalar potential (3.5). As F = ρB, the covariant
ij ρϕ
brackets are given by
∂Q ∂K ∂K ∂Q ∂K ∂Q
{K,Q} = D K − D Q+eBρ − , (3.8)
i i
∂π ∂π ∂π ∂π ∂π ∂π
i i (cid:18) ρ ϕ ϕ ρ(cid:19)
where the covariant derivatives D are defined by (2.3) with non-zero connection coefficients
i
Γ ϕ = 1/ρ, Γ ρ = −ρ. As the angle ϕ is a cyclic co-ordinate, the corresponding canonical
ρϕ ϕϕ
momentum is conserved. In the present covariant formalism this follows from the existence
of a Killing vector and a Killing scalar, C(1)ρ,C(1)z,C(1)ϕ = (0,0,1), C(0) = ω ρ2, re-
L
spectively, where ω = eB/2. They comb(cid:0)ine into the consta(cid:1)nt of the motion, namely the
L
z-component of the total angular momentum,
L = C(1)iπ +C(0) = π +ω ρ2. (3.9)
z i ϕ L
The Hamilton equations dξi/dt = {ξi,H} = gijπ imply dϕ/dt = π /ρ2 = L /ρ2−ω ; ω is
j ϕ z L L
hence the Larmor frequency.
As discussed in [16–19], the model allows for more constants of motion whenever certain
specific conditions on the frequencies hold, namely for certain exceptional values of τ =
ω / ω2 +ω2 .
z 0 L
•pWe first observe that there is a rank-2 Killing tensor
2z −ρ 0
1 z
C(2) =  −ρ 0 0  ⇔ C(2)ijπ π = zπ2 −ρπ π + π2. (3.10)
ij 2 i j ρ ρ z ρ2 ϕ
 0 0 2ρ2z 
 
 
 
Then the generalized Killing equation (2.9) for C(1) is solved by
C(1) = 0,0,eBρ2z ⇔ C(1)iπ = eBzπ . (3.11)
i i ϕ
(cid:0) (cid:1)
5
This vector is orthogonal to the gradient of Φ, as required by the first equation in (2.7).
Finally, the Killing scalar C(0) must satisfy
aρz
C(0) = 4ω2 +2ω2 −ω2 ρz + ,
,ρ L 0 z (ρ2 +z2)3/2
(3.12)
(cid:0) aρ2(cid:1)
C(0) = −ω2ρ2 − , C(0) = 0.
,z ρ (ρ2 +z2)3/2 ,ϕ
A solution, namely
az
C(0) = −ω2ρ2z − , (3.13)
ρ
ρ2 +z2
exists, provided the frequencies satisfy
p
1
ω2 +ω2 = ω2. (3.14)
0 L 4 z
Combining the results, the full constant of motion,
z az
Q = zπ2 −ρπ π + π2 +2ω zπ −ω2ρ2z − , (3.15)
1 ρ ρ z ρ2 ϕ L ϕ 0 ρ2 +z2
We recover hence the Runge-Lenz-type quadratic expression found before for τ = 2, when
p
the system is separable in parabolic coordinates [16–19].
Another constant can be constructed starting from the rank-4 Killing tensor
1 1 z2
C(4)ijklπ π π π = ρ2π4 −2ρzπ π3 +z2π2π2 + π4 +π2π2 + 2+ π2π2. (3.16)
4! i j k l z ρ z ρ z ρ2 ϕ ρ ϕ ρ2 z ϕ
(cid:18) (cid:19)
Then solving eqs. (2.9) for C(3) one finds the minimal solution
1
C(3)ijkπ π π = 2ω π ρ2π2 +(2ρ2 +z2)π2 , (3.17)
3! i j k L ϕ ρ z
(cid:0) (cid:1)
modulo a rather long list of separate rank-3 Killing tensor expressions, defining independent
constants of motion. These are discussed in eq. (3.25) below.
The terms quadratic in the covariant momenta are now obtained in a straightforward
way by requiring all contributions of order π3 to the bracket {Q,H} to cancel; this gives the
minimal expression,
1 2aρ2
C(2)ijπ π = (2ω2 −ω2)z2ρ2 +2ω2ρ4 − π2
2 i j " z 0 L z2 +ρ2# z
2 p 2azρ
+ 2ω2−5ω2 +2ω2 z3ρ+ π π
"3 0 z L z2 +ρ2# z ρ
(cid:0) (cid:1)
1
+ ω2ρ4 − ω2−4ω2 +ω2 zp4 π2
L 3 0 z L ρ
(cid:20) (cid:21)
(cid:0) (cid:1)
1 z4 2a
+ 2ω2z2 + ω2 −5ω2 ρ2 − ω2 −4ω2+ω2 − π2.
" z 0 L 3 0 z L ρ2 z2 +ρ2# ϕ
(cid:0) (cid:1) (cid:0) (cid:1) (3.18)
p
6
Calculating the contributions of order π2, we have to add a linear term
1 2aρ2
C(1)iπ =−2ω π ω2 −4ω2 +ω2 z4 −2ω2z2ρ2 + 3ω2 −ω2 ρ4 + .
i L ϕ"3 0 z L z L 0 z2 +ρ2#
(cid:0) (cid:1) (cid:0) (cid:1) (3.19)
p
It remains to find a C(0)(z,ρ) such that the bracket closes. This requires
16 4 4 2a2zρ2
C(0) = 4ω4− ω2ω2 + ω4 + ω2ω2 z3ρ2 +4ω2ω2zρ4 −
,z z 3 z ρ 3 0 3 L 0 L z (z2 +ρ2)2
(cid:18) (cid:19)
a 10 4 4
+ − ω2 + ω2 + ω2 z3ρ2 + −4ω2 +2ω2 +4ω2 zρ4 ,
(z2 +ρ2)3/2 3 z 3 0 3 L z 0 L
(cid:20)(cid:18) (cid:19) (cid:21)
(cid:0) (cid:1)
10 2 20 4
C(0) = − ω4 +4ω2ω2 − ω4 + ω2ω2 −2ω2ω2 − ω4 z4ρ
,ρ 3 z z 0 3 0 3 L z L 0 3 L
(cid:18) (cid:19)
2a2z2ρ
+ 8ω2ω2z2ρ3 −6ω2 2ω2 −ω2 ρ5 +
L z L L 0 (z2 +ρ2)2
a 2 (cid:0) (cid:1)
+ ω2 +2ω2 +ω2 z4ρ+2 ω2 −4ω2 z2ρ3 −6ω2ρ5 .
(z2 +ρ2)3/2 3 0 z L z L L
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1) (3.20)
Let us first turn off the Coulomb potential, i.e., consider the a-independent part of these
eqns. Then the integrability condition is
2ω2 −5ω2 +2ω2 2 = 9ω4, (3.21)
L z 0 z
(cid:0) (cid:1)
with allows for two solutions, namely
(a) ω2 = 4ω2 −ω2, (b) ω2 = ω2 −ω2. (3.22)
L z 0 L z 0
Thus, for the magnetic problem with a confining harmonic potential but with no Coulomb
potential, there are two values of the Larmor frequency ω for which there is a quartic
L
constant of motion.
•Incontrast, whena 6= 0,i.e., whentheCoulomb potential is switched on,thetermslinear
in a are integrable only if condition (a) is satisfied (terms proportional to a2 are actually
always integrable) – which is, indeed, the integrable but non-separable case τ = 1/2 in [19]
cf. [16–19]. Imposing condition (a), the minimal solution for C(0) becomes
C(0) = ω4z4ρ2 +2ω2ω2z2ρ4 −ω2 3ω2 −4ω2 ρ6
z z L L L z
2a (cid:0) a2z2 −(cid:1) ρ2 (3.23)
+ ω2z2ρ2 −ω2ρ4 − ,
z2 +ρ2 z L 2 z2 +ρ2
(cid:2) (cid:3)
p
7
with ω2+ω2 = 4ω2. Then the sum of the expressions (3.16), (3.17), (3.18), (3.19) and (3.23)
L 0 z
represents, for this special value of the magnetic field, the quartic constant of the motion in
the CM system. Explicitly,
1 z2
Q = ρ2π4 −2ρzπ π3 +z2π2π2 + π4 +π2π2 + 2+ π2π2
2 z ρ z ρ z ρ2 ϕ ρ ϕ ρ2 z ϕ
(cid:18) (cid:19)
2aρ2
+2ω π ρ2π2 +(2ρ2 +z2)π2 + (2ω2 −ω2)z2ρ2 +2ω2ρ4 − π2
L ϕ ρ z z 0 L z
" z2 +ρ2#
(cid:0) (cid:1)
2 2azρ 1
+ 2ω2−5ω2 +2ω2 z3ρ+ π π + ω2ρ4 − pω2 −4ω2 +ω2 z4 π2
"3 0 z L z2 +ρ2# z ρ L 3 0 z L ρ
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
1 p z4 2a
+ 2ω2z2 + ω2 −5ω2 ρ2 − ω2 −4ω2+ω2 − π2
" z 0 L 3 0 z L ρ2 z2 +ρ2# ϕ
(cid:0) (cid:1) (cid:0) (cid:1)
1 2aρ2
−2ω π ω2 −4ω2+ω2 z4 −2ω2z2ρ2 + 3ω2 −ωp2 ρ4 +
L ϕ"3 0 z L z L 0 z2 +ρ2#
(cid:0) (cid:1) (cid:0) 2a (cid:1) a2z2 −ρ2
+ω4z4ρ2 +2ω2ω2z2ρ4 −ω2 3ω2 −4ω2 ρ6+ ω2z2ρp2 −ω2ρ4 − ,
z z L L L z z2 +ρ2 z L 2 z2 +ρ2
(cid:0) (cid:1) (cid:2) (cid:3) (3.24)
p
which is in fact the conserved quantity found in the integrable-but-non-separable case τ =
1/2 using quite different methods [19]. By construction, the coefficients of the quartic term
(3.16) define a rank-4 Killing tensor w.r.t. the metric (3.7).
More generally, the complete list of rank-3 Killing tensors is
K(3) = π3, K(3) = π3,
1 z 2 ϕ
1
K(3) = π π2, K(3) = π π2 + π2 ,
3 z ϕ 4 z ρ ρ2 ϕ
(cid:18) (cid:19)
1 1 (3.25)
K(3) = π π2 + π2 , K(3) = π −ρπ π +z π2 + π2 ,
5 ϕ ρ ρ2 ϕ 6 z ρ z ρ ρ2 ϕ
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
1 1 1
K(3) = π ρ2π2 −zρπ π + z2 π2 + π2 .
7 z 2 z ρ z 2 ρ ρ2 ϕ
(cid:20) (cid:18) (cid:19)(cid:21)
Note, however, that these are composed of direct products of lower-rank Killing tensors
(1) (1)
and vectors. In particular, K = π and K = π define Killing vectors by themselves:
z z ϕ ϕ
(1)i (1)i
K = (0,1,0), K = (0,0,1).
z ϕ
The expressions in eqs. (3.25) are products of these Killing vectors with each other and
with the rank-2 Killing tensors
1 1
K(2) = π2 + π2, K(2) = −ρπ π +z π2 + π2 ,
1 ρ ρ2 ϕ 2 ρ z ρ ρ2 ϕ
(cid:18) (cid:19) (3.26)
1 1 1
K(2) = ρ2π2 −zρπ π + z2 π2 + π2 .
3 2 z ρ z 2 ρ ρ2 ϕ
(cid:18) (cid:19)
8
(2)
We discuss these expressions in turn. First, K satisfies the bracket relation
1
a
K(2),H = −2ρπ ω2+ . (3.27)
1 ρ 0 (z2 +ρ2)3/2
(cid:18) (cid:19)
n o
(2)
Now as π is not a Killing vector, K can be turned into a constant of motion only by
ρ 1
(0)
adding a scalar term K such that
1
a
K(0) = 0, K(0) = 2ρ ω2 + . (3.28)
1,z 1,ρ 0 (z2 +ρ2)3/2
(cid:18) (cid:19)
The solution of the 2nd equation:
a
K(0) = ω2ρ2 − , (3.29)
1 0
z2 +ρ2
p
satisfies the first equation (3.28) only if a = 0. Therefore this quadratic Killing tensor
generates a constant of motion only in the absence of a Coulomb potential: a = 0.
An alternative is to replace the 3D Coulomb potential by a 2D one:
1 1 a
Φ → Φ˜ = ω2z2 + ω2ρ2 − . (3.30)
2 z 2 0 ρ
(2)
Next, observe that K is identical to the Killing tensor (3.10). We have already seen that
2
it can be extended to a constant of motion only if the Larmor frequency is tuned to take
the value (3.14).
(2)
Finally, we discuss K . A straightforward calculation along the previous lines shows
3
that it can also be extended to a complete constant of motion,
1 1 1 1
Q = ρ2π2 −zρπ π + z2 π2 + π2 + ω z2π + ω2z2ρ2, (3.31)
3 2 z z ρ 2 ρ ρ2 ϕ L ϕ 2 L
(cid:18) (cid:19)
provided ω2 = ω2 −ω2, — which is condition (b) in (3.22).
L z 0
We observe that (3.31) is in fact the difference of two separately conserved quantities
found in Ref. [19], 1(L2 − L2), i.e., (half of) the total angular momentum squared, L2,
2 z
minus L2, the square of the third component of L . This is no surprise since condition (b)
z z
in (3.22) means τ = 1, which amounts to spherical symmetry and hence conserved total
angular momentum after elimination of the magnetic field [19].
9
4. ALGEBRAIC STRUCTURE
Whenwehaveseveral symmetries, theiralgebraicstructureisoffundamental importance.
(Remembertheo(4)/o(3,1)dynamicalsymmetryoftheKeplerproblem.) HowdotheKilling
tensors reflect the Poisson algebra structure of the associated conserved quantities?
Theanswer isnon-trivialletaloneinthesimplest, rank-1case, whenthePoissonstructure
of the conserved quantities may not be the same as that of the generating vectors under Lie
bracket. Just consider a free particle: while the vectors of the infinitesimal action on space-
time span the center-less Galilei Lie algebra, the associated conserved quantities realize its
central extension (called the Bargmann algebra).
This problem can be conveniently dealt with using a higher-dimensional framework [20]
and Schouten-Nijenhuis algebras [12]. Full details will be presented elsewhere. Here we
satisfy ourselves with some general remarks about the bracket algebra [21]. Let J(p) be
constructed from a highest-rank Killing tensor of rank p. By construction, the bracket of
two such constants of motion has the general structure
{J(p),J(q)} ∼ J(p+q 1). (4.1)
−
It follows that the generators of rank p = 1 form a closed algebra, namely a Lie algebra if the
structurefunctionsareconstant. Theconstantsofrankp ≥ 1thenmustformrepresentations
of this algebra:
{J(p),J(1)} ∼ J(p). (4.2)
If there is more than one constant of the motion of rank p,q ≥ 2, their bracket generates
constants of motion of higher rank p + q − 1 ≥ p + 1. Therefore either the J(p), p ≥ 2,
form a trivial representation of the Lie algebra and all their brackets vanish, or an infinite-
dimensional set of constants of the motion is generated. Well-known examples of such
infinite-dimensional algebras are the Virasoro and Kac-Moody algebras. However, these
non-trivial infinite-dimensional algebras arise only for infinite-dimensional systems. In the
finite-dimensional case we expect the brackets of higher-rank constants of motion to vanish,
or represent simple powers and products of lower-rank constants.
Obviously, the Lie-algebra of constants J(1) generates transformations in configuration
space, and corresponding configuration-dependent linear transformations in momentum
10